Multivariable Calculus (Calc III) - Complete Semester Course - Quadric Surfaces, Multivariable Calculus

In this lecture, we look at quadric surfaces and practice graphing equations in (x,y,z)-space using cross sections. To lay the groundwork, we review four conic sections in the xy-plane: a circle, an ellipse, a parabola, and a hyperbola. Understanding these conic sections is crucial for grasping quadric surfaces. The Paraboloid 𝑧=𝑥^2+𝑦^2. We examine the equation 𝑧=𝑥^2+𝑦^2 and approach its graphing through cross-sections. For 𝑧=0, the equation simplifies to the origin. As 𝑧 increases, we observe circles with increasing radii in the respective 𝑧-planes. Additionally, considering the cross-sections 𝑥=0 and 𝑦=0, we find parabolas in the yz-plane and xz-plane, respectively. This leads us to conclude that the graph is a paraboloid. We discuss variations of the paraboloid, such as elliptic paraboloids (e.g., 𝑧=2𝑥^2+3𝑦^2) and paraboloids opening downwards (e.g., 𝑧=−𝑥^2−𝑦^2). Next, we graph 𝑧=sqrt(𝑥2+𝑦2) using similar cross-sectional methods. The cross-sections for 𝑧 yield circles with radii equal to 𝑧, while the cross-sections 𝑥=0 and 𝑦=0 produce V-shaped graphs in the yz-plane and xz-plane. This shape is identified as a cone, specifically a single cone. Double Cone 𝑧^2=𝑥^2+𝑦^2: We then consider 𝑧^2=𝑥^2+𝑦^2, which generalizes the previous example to include both positive and negative 𝑧 values, forming a double cone. Hyperbolic Paraboloid: We explore the hyperbolic paraboloid 𝑧=𝑥^2−𝑦^2 through cross-sections in 𝑥 and 𝑦. The cross-sections in 𝑥 yield downward-opening parabolas, while those in 𝑦 yield upward-opening parabolas. This forms a saddle-like shape, also known as a hyperbolic paraboloid. Hyperboloid of One Sheet 𝑥^2+𝑦^2−𝑧^2=1: Finally, we graph 𝑥^2+𝑦^2−𝑧^2=1, which results in hyperbolas for both 𝑥 and 𝑦 cross-sections and a circle for 𝑧=0. This structure is identified as a hyperboloid of one sheet. Multivariable Calculus Unit 3 Lecture 2. (Previous video: https://youtu.be/blbBzLB4xqs) #mathematics #multivariablecalculus #calculus #conicsection #conicsections #iitjammathematics #calculus3